A subspace of ${\mathbb{F}}_2^n$ is called cyclically covering if every vector in ${\mathbb{F}}_2^n$ has a cyclic shift which is inside the subspace. Let $h_2(n)$ denote the largest possible codimension of a cyclically covering subspace of ${\mathbb{F}}_2^n$. We show that $h_2(p)=2$ for every prime p such that 2 is a primitive root modulo p, which, assuming Artin's conjecture, answers a question of Peter Cameron from 1991. We also prove various bounds on $h_2(ab)$ depending on $h_2(a)$ and $h_2(b)$ and extend some of our results to a more general set-up proposed by Cameron, Ellis and Raynaud.

中文翻译：

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循环覆盖中的子空间 ${\mathbb{F}}_2^{ñ}$

的子空间 ${\mathbb{F}}_2^{ñ}$ 如果每个向量都被称为循环覆盖 ${\mathbb{F}}_2^{ñ}$具有在子空间内的循环移位。让$H_2（ñ）$ 表示的周期覆盖子空间的最大可能维数 ${\mathbb{F}}_2^{ñ}$。我们证明$H_2（p）=2$对于每个素数p，使得2是原始根模p，假设阿丁的猜想，它回答了1991年彼得·卡梅隆的问题。我们还证明了$H_2（\mathrm{一种}b）$ 根据 $H_2（\mathrm{一种}）$ 和 $H_2（b）$ 并将我们的一些结果扩展到Cameron，Ellis和Raynaud提出的更一般的设置。